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I have a DiskStation DS214play using DSM 5.1-5022 Update 3 and iTunes Server version 1.3-2402

Thinkpad (Windows): I open iTunes and cannot see the DiskStation shared library. So, I open the DiskStation desktop within my browser and stop/start the iTunes server. I can now see the DiskStation within iTunes

MacBook Pro: I open iTunes and cannot see the DiskStation shared library. So, I open the DiskStation desktop within my browser and stop/start the iTunes server. I can now see the DiskStation within iTunes

I now go back to my Thinkpad and the DiskStation share has now gone If I stop/start the server, the share returns... but the Macbook then looses its connection with the share

The only way of reconnecting the shared library is by stop/starting the server, but means that only one device can access the server at a time.

Have you tried uninstalling iTunes Server, rebooting the NAS, reinstalling and then rebooting the NAS again, then test? Do you have any Firewall settings within the NAS, router or network affecting this?